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1: Functions of Several Variables

1: Functions of Several Variables

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97817_14_ch14_p901-909.qk_97817_14_ch14_p901-909 11/8/10 1:26 PM Page 903

SECTION 14.1

y

x=1

0

x

Since ln͑ y 2 Ϫ x͒ is defined only when y 2 Ϫ x Ͼ 0, that is, x Ͻ y 2, the domain of f is

D ෇ ͕͑x, y͒ x Ͻ y 2 ͖. This is the set of points to the left of the parabola x ෇ y 2. (See

Figure 3.)

Խ

Not all functions can be represented by explicit formulas. The function in the next example is described verbally and by numerical estimates of its values.

FIGURE 2

œ„„„„„„„

x+y+1

x-1

EXAMPLE 2 In regions with severe winter weather, the wind-chill index is often used to

describe the apparent severity of the cold. This index W is a subjective temperature that

depends on the actual temperature T and the wind speed v. So W is a function of T and v,

and we can write W ෇ f ͑T, v͒. Table 1 records values of W compiled by the National

Weather Service of the US and the Meteorological Service of Canada.

y

x=¥

0

f ͑3, 2͒ ෇ 3 ln͑2 2 Ϫ 3͒ ෇ 3 ln 1 ෇ 0

(b)

_1

Domain of f(x, y)=

903

the line y ෇ Ϫx Ϫ 1, while x 1 means that the points on the line x ෇ 1 must be

excluded from the domain. (See Figure 2.)

x+y+1=0

_1

FUNCTIONS OF SEVERAL VARIABLES

x

TABLE 1 Wind-chill index as a function of air temperature and wind speed

Wind speed (km/h)

T

FIGURE 3

The New Wind-Chill Index

A new wind-chill index was introduced in

November of 2001 and is more accurate than

the old index for measuring how cold it feels

when it’s windy. The new index is based on a

model of how fast a human face loses heat. It

was developed through clinical trials in which

volunteers were exposed to a variety of temperatures and wind speeds in a refrigerated wind

tunnel.

Actual temperature (°C)

Domain of f(x, y)=x ln(¥-x)

v

10

5

15

20

25

30

40

50

60

70

80

5

4

3

2

1

1

0

Ϫ1

Ϫ1

Ϫ2

Ϫ2

Ϫ3

0

Ϫ2

Ϫ3

Ϫ4

Ϫ5

Ϫ6

Ϫ6

Ϫ7

Ϫ8

Ϫ9

Ϫ9

Ϫ10

Ϫ5

Ϫ7

Ϫ9

Ϫ11

Ϫ12

Ϫ12

Ϫ13

Ϫ14

Ϫ15

Ϫ16

Ϫ16

Ϫ17

Ϫ10

Ϫ13

Ϫ15

Ϫ17

Ϫ18

Ϫ19

Ϫ20

Ϫ21

Ϫ22

Ϫ23

Ϫ23

Ϫ24

Ϫ15

Ϫ19

Ϫ21

Ϫ23

Ϫ24

Ϫ25

Ϫ26

Ϫ27

Ϫ29

Ϫ30

Ϫ30

Ϫ31

Ϫ20

Ϫ24

Ϫ27

Ϫ29

Ϫ30

Ϫ32

Ϫ33

Ϫ34

Ϫ35

Ϫ36

Ϫ37

Ϫ38

Ϫ25

Ϫ30

Ϫ33

Ϫ35

Ϫ37

Ϫ38

Ϫ39

Ϫ41

Ϫ42

Ϫ43

Ϫ44

Ϫ45

Ϫ30

Ϫ36

Ϫ39

Ϫ41

Ϫ43

Ϫ44

Ϫ46

Ϫ48

Ϫ49

Ϫ50

Ϫ51

Ϫ52

Ϫ35

Ϫ41

Ϫ45

Ϫ48

Ϫ49

Ϫ51

Ϫ52

Ϫ54

Ϫ56

Ϫ57

Ϫ58

Ϫ60

Ϫ40

Ϫ47

Ϫ51

Ϫ54

Ϫ56

Ϫ57

Ϫ59

Ϫ61

Ϫ63

Ϫ64

Ϫ65

Ϫ67

For instance, the table shows that if the temperature is Ϫ5ЊC and the wind speed is

50 km͞h, then subjectively it would feel as cold as a temperature of about Ϫ15ЊC with

no wind. So

f ͑Ϫ5, 50͒ ෇ Ϫ15

EXAMPLE 3 In 1928 Charles Cobb and Paul Douglas published a study in which they

modeled the growth of the American economy during the period 1899–1922. They considered a simplified view of the economy in which production output is determined by the

amount of labor involved and the amount of capital invested. While there are many other

factors affecting economic performance, their model proved to be remarkably accurate.

The function they used to model production was of the form

1

P͑L, K͒ ෇ bL␣K 1Ϫ␣

where P is the total production (the monetary value of all goods produced in a year),

L is the amount of labor (the total number of person-hours worked in a year), and K is

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904

PARTIAL DERIVATIVES

CHAPTER 14

the amount of capital invested (the monetary worth of all machinery, equipment, and

buildings). In Section 14.3 we will show how the form of Equation 1 follows from certain economic assumptions.

Cobb and Douglas used economic data published by the government to obtain

Table 2. They took the year 1899 as a baseline and P, L, and K for 1899 were each

assigned the value 100. The values for other years were expressed as percentages of

the 1899 figures.

Cobb and Douglas used the method of least squares to fit the data of Table 2 to the

function

TABLE 2

.

Year

P

L

K

1899

1900

1901

1902

1903

1904

1905

1906

1907

1908

1909

1910

1911

1912

1913

1914

1915

1916

1917

1918

1919

1920

1921

1922

100

101

112

122

124

122

143

152

151

126

155

159

153

177

184

169

189

225

227

223

218

231

179

240

100

105

110

117

122

121

125

134

140

123

143

147

148

155

156

152

156

183

198

201

196

194

146

161

100

107

114

122

131

138

149

163

176

185

198

208

216

226

236

244

266

298

335

366

387

407

417

431

P͑L, K͒ ෇ 1.01L0.75K 0.25

2

(See Exercise 79 for the details.)

If we use the model given by the function in Equation 2 to compute the production in

the years 1910 and 1920, we get the values

P͑147, 208͒ ෇ 1.01͑147͒0.75͑208͒0.25 Ϸ 161.9

P͑194, 407͒ ෇ 1.01͑194͒0.75͑407͒0.25 Ϸ 235.8

which are quite close to the actual values, 159 and 231.

The production function 1 has subsequently been used in many settings, ranging

from individual firms to global economics. It has become known as the Cobb-Douglas

production function. Its domain is ͕͑L, K͒ L ജ 0, K ജ 0͖ because L and K represent

labor and capital and are therefore never negative.

Խ

EXAMPLE 4 Find the domain and range of t͑x, y͒ ෇ s9 Ϫ x 2 Ϫ y 2 .

SOLUTION The domain of t is

y

D ෇ ͕͑x, y͒

≈+¥=9

Խ 9Ϫx

2

Ϫ y 2 ജ 0͖ ෇ ͕͑x, y͒

Խx

2

ϩ y 2 ഛ 9͖

which is the disk with center ͑0, 0͒ and radius 3. (See Figure 4.) The range of t is

_3

3

͕ z Խ z ෇ s9 Ϫ x 2 Ϫ y 2 , ͑x, y͒ ʦ D͖

x

Since z is a positive square root, z ജ 0. Also, because 9 Ϫ x 2 Ϫ y 2 ഛ 9, we have

s9 Ϫ x 2 Ϫ y 2 ഛ 3

So the range is

FIGURE 4

͕z

9-≈-¥

Domain of g(x, y)=œ„„„„„„„„„

Խ 0 ഛ z ഛ 3͖ ෇ ͓0, 3͔

Graphs

z

S

{ x, y, f (x, y)}

Another way of visualizing the behavior of a function of two variables is to consider its

graph.

Definition If f is a function of two variables with domain D, then the graph of f

is the set of all points ͑x, y, z͒ in ‫ޒ‬3 such that z ෇ f ͑x, y͒ and ͑x, y͒ is in D.

f(x, y)

0

D

x

FIGURE 5

(x, y, 0)

y

Just as the graph of a function f of one variable is a curve C with equation y ෇ f ͑x͒, so

the graph of a function f of two variables is a surface S with equation z ෇ f ͑x, y͒. We can

visualize the graph S of f as lying directly above or below its domain D in the xy-plane (see

Figure 5).

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97817_14_ch14_p901-909.qk_97817_14_ch14_p901-909 11/8/10 1:26 PM Page 905

SECTION 14.1

z

FUNCTIONS OF SEVERAL VARIABLES

905

EXAMPLE 5 Sketch the graph of the function f ͑x, y͒ ෇ 6 Ϫ 3x Ϫ 2y.

(0, 0, 6)

SOLUTION The graph of f has the equation z ෇ 6 Ϫ 3x Ϫ 2y, or 3x ϩ 2y ϩ z ෇ 6,

which represents a plane. To graph the plane we first find the intercepts. Putting

y ෇ z ෇ 0 in the equation, we get x ෇ 2 as the x-intercept. Similarly, the y-intercept is 3

and the z-intercept is 6. This helps us sketch the portion of the graph that lies in the first

octant in Figure 6.

(0, 3, 0)

(2, 0, 0)

y

The function in Example 5 is a special case of the function

x

f ͑x, y͒ ෇ ax ϩ by ϩ c

FIGURE 6

which is called a linear function. The graph of such a function has the equation

z ෇ ax ϩ by ϩ c

or

ax ϩ by Ϫ z ϩ c ෇ 0

so it is a plane. In much the same way that linear functions of one variable are important in

single-variable calculus, we will see that linear functions of two variables play a central

role in multivariable calculus.

z

0

(3, 0, 0)

v

(0, 0, 3)

EXAMPLE 6 Sketch the graph of t͑x, y͒ ෇ s9 Ϫ x 2 Ϫ y 2 .

SOLUTION The graph has equation z ෇ s9 Ϫ x 2 Ϫ y 2 . We square both sides of this

equation to obtain z 2 ෇ 9 Ϫ x 2 Ϫ y 2, or x 2 ϩ y 2 ϩ z 2 ෇ 9, which we recognize as an

equation of the sphere with center the origin and radius 3. But, since z ജ 0, the graph of

t is just the top half of this sphere (see Figure 7).

(0, 3, 0)

y

x

FIGURE 7

Graph of g(x, y)=œ„„„„„„„„„

9-≈-¥

NOTE An entire sphere can’t be represented by a single function of x and y. As we saw

in Example 6, the upper hemisphere of the sphere x 2 ϩ y 2 ϩ z 2 ෇ 9 is represented by the

function t͑x, y͒ ෇ s9 Ϫ x 2 Ϫ y 2 . The lower hemisphere is represented by the function

h͑x, y͒ ෇ Ϫs9 Ϫ x 2 Ϫ y 2 .

EXAMPLE 7 Use a computer to draw the graph of the Cobb-Douglas production function

P͑L, K͒ ෇ 1.01L0.75K 0.25.

SOLUTION Figure 8 shows the graph of P for values of the labor L and capital K that lie

between 0 and 300. The computer has drawn the surface by plotting vertical traces. We

see from these traces that the value of the production P increases as either L or K

increases, as is to be expected.

300

200

P

100

0

300

FIGURE 8

v

200

100

K

0 0

100

200

300

L

EXAMPLE 8 Find the domain and range and sketch the graph of h͑x, y͒ ෇ 4x 2 ϩ y 2.

SOLUTION Notice that h͑x, y͒ is defined for all possible ordered pairs of real numbers

͑x, y͒, so the domain is ‫ޒ‬2, the entire xy-plane. The range of h is the set ͓0, ϱ͒ of all nonnegative real numbers. [Notice that x 2 ജ 0 and y 2 ജ 0, so h͑x, y͒ ജ 0 for all x and y.]

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

97817_14_ch14_p901-909.qk_97817_14_ch14_p901-909 11/8/10 1:26 PM Page 906

906

CHAPTER 14

PARTIAL DERIVATIVES

The graph of h has the equation z ෇ 4x 2 ϩ y 2, which is the elliptic paraboloid that

we sketched in Example 4 in Section 12.6. Horizontal traces are ellipses and vertical

traces are parabolas (see Figure 9).

z

FIGURE 9

x

Graph of h(x, y)=4≈+¥

y

Computer programs are readily available for graphing functions of two variables. In most

such programs, traces in the vertical planes x ෇ k and y ෇ k are drawn for equally spaced

values of k and parts of the graph are eliminated using hidden line removal.

Figure 10 shows computer-generated graphs of several functions. Notice that we get an

especially good picture of a function when rotation is used to give views from different

z

z

x

y

x

(b) f(x, y)=(≈+3¥)e _≈_¥

(a) f(x, y)=(≈+3¥)e _≈_¥

z

z

x

y

x

(c) f(x, y)=sin x+sin y

y

(d) f(x, y)=

sin x  sin y

xy

FIGURE 10

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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SECTION 14.1

FUNCTIONS OF SEVERAL VARIABLES

907

vantage points. In parts (a) and (b) the graph of f is very flat and close to the xy-plane except

near the origin; this is because eϪx Ϫ y is very small when x or y is large.

2

2

Level Curves

So far we have two methods for visualizing functions: arrow diagrams and graphs. A third

method, borrowed from mapmakers, is a contour map on which points of constant elevation

are joined to form contour lines, or level curves.

Definition The level curves of a function f of two variables are the curves with

equations f ͑x, y͒ ෇ k, where k is a constant (in the range of f ).

A level curve f ͑x, y͒ ෇ k is the set of all points in the domain of f at which f takes on

a given value k. In other words, it shows where the graph of f has height k.

You can see from Figure 11 the relation between level curves and horizontal traces. The

level curves f ͑x, y͒ ෇ k are just the traces of the graph of f in the horizontal plane

z ෇ k projected down to the xy-plane. So if you draw the level curves of a function and

visualize them being lifted up to the surface at the indicated height, then you can mentally

piece together a picture of the graph. The surface is steep where the level curves are close

together. It is somewhat flatter where they are farther apart.

z

40

45

00

45

00

50

00

LONESOME MTN.

0

A

55

00

B

y

50

x

TEC Visual 14.1A animates Figure 11 by

showing level curves being lifted up to graphs

of functions.

0

FIGURE 11

450

f(x, y)=20

00

k=45

k=40

k=35

k=30

k=25

k=20

Lon

eso

me

Cree

k

FIGURE 12

One common example of level curves occurs in topographic maps of mountainous

regions, such as the map in Figure 12. The level curves are curves of constant elevation

above sea level. If you walk along one of these contour lines, you neither ascend nor descend.

Another common example is the temperature function introduced in the opening paragraph

of this section. Here the level curves are called isothermals and join locations with the same

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

97817_14_ch14_p901-909.qk_97817_14_ch14_p901-909 11/8/10 1:26 PM Page 908

908

CHAPTER 14

PARTIAL DERIVATIVES

temperature. Figure 13 shows a weather map of the world indicating the average January

temperatures. The isothermals are the curves that separate the colored bands.

FIGURE 13

World mean sea-level temperatures

in January in degrees Celsius

From Atmosphere: Introduction to Meteorology, 4th Edition, 1989.

© 1989 Pearson Education, Inc.

y

EXAMPLE 9 A contour map for a function f is shown in Figure 14. Use it to estimate the

values of f ͑1, 3͒ and f ͑4, 5͒.

50

5

SOLUTION The point (1, 3) lies partway between the level curves with z-values 70

4

and 80. We estimate that

3

f ͑1, 3͒ Ϸ 73

2

1

0

1

80

70

60

50

2

3

80

70

60

4

Similarly, we estimate that

5

x

FIGURE 14

f ͑4, 5͒ Ϸ 56

EXAMPLE 10 Sketch the level curves of the function f ͑x, y͒ ෇ 6 Ϫ 3x Ϫ 2y for the

values k ෇ Ϫ6, 0, 6, 12.

SOLUTION The level curves are

6 Ϫ 3x Ϫ 2y ෇ k

3x ϩ 2y ϩ ͑k Ϫ 6͒ ෇ 0

or

This is a family of lines with slope Ϫ 32 . The four particular level curves with

k ෇ Ϫ6, 0, 6, and 12 are 3x ϩ 2y Ϫ 12 ෇ 0, 3x ϩ 2y Ϫ 6 ෇ 0, 3x ϩ 2y ෇ 0, and

3x ϩ 2y ϩ 6 ෇ 0. They are sketched in Figure 15. The level curves are equally spaced

parallel lines because the graph of f is a plane (see Figure 6).

y

0

_6

k=

0

k=

6

k=

12

k=

FIGURE 15

Contour map of

f(x, y)=6-3x-2y

x

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97817_14_ch14_p901-909.qk_97817_14_ch14_p901-909 11/8/10 1:26 PM Page 909

FUNCTIONS OF SEVERAL VARIABLES

SECTION 14.1

v

909

EXAMPLE 11 Sketch the level curves of the function

t͑x, y͒ ෇ s9 Ϫ x 2 Ϫ y 2

k ෇ 0, 1, 2, 3

for

SOLUTION The level curves are

s9 Ϫ x 2 Ϫ y 2 ෇ k

x2 ϩ y2 ෇ 9 Ϫ k2

or

This is a family of concentric circles with center ͑0, 0͒ and radius s9 Ϫ k 2 . The cases

k ෇ 0, 1, 2, 3 are shown in Figure 16. Try to visualize these level curves lifted up to

form a surface and compare with the graph of t (a hemisphere) in Figure 7. (See TEC

Visual 14.1A.)

y

k=3

k=2

k=1

k=0

(3, 0)

0

x

FIGURE 16

Contour map of g(x, y)=œ„„„„„„„„„

9-≈-¥

EXAMPLE 12 Sketch some level curves of the function h͑x, y͒ ෇ 4x 2 ϩ y 2 ϩ 1.

SOLUTION The level curves are

4x 2 ϩ y 2 ϩ 1 ෇ k

or

1

4

x2

y2

ϩ

෇1

͑k Ϫ 1͒

kϪ1

which, for k Ͼ 1, describes a family of ellipses with semiaxes 12 sk Ϫ 1 and sk Ϫ 1 .

Figure 17(a) shows a contour map of h drawn by a computer. Figure 17(b) shows these

level curves lifted up to the graph of h (an elliptic paraboloid) where they become horizontal traces. We see from Figure 17 how the graph of h is put together from the level

curves.

y

z

TEC Visual 14.1B demonstrates the

connection between surfaces and their

contour maps.

x

x

FIGURE 17

The graph of h(x, y)=4≈+¥+1

is formed by lifting the level curves.

y

(a) Contour map

(b) Horizontal traces are raised level curves

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

97817_14_ch14_p910-919.qk_97817_14_ch14_p910-919 11/8/10 1:27 PM Page 910

910

CHAPTER 14

PARTIAL DERIVATIVES

K

EXAMPLE 13 Plot level curves for the Cobb-Douglas production function of Example 3.

300

SOLUTION In Figure 18 we use a computer to draw a contour plot for the Cobb-

Douglas production function

P͑L, K͒ ෇ 1.01L 0.75K 0.25

200

Level curves are labeled with the value of the production P. For instance, the level curve

labeled 140 shows all values of the labor L and capital investment K that result in a production of P ෇ 140. We see that, for a fixed value of P, as L increases K decreases, and

vice versa.

220

180

100

140

100

100

200

300 L

FIGURE 18

For some purposes, a contour map is more useful than a graph. That is certainly true in

Example 13. (Compare Figure 18 with Figure 8.) It is also true in estimating function values, as in Example 9.

Figure 19 shows some computer-generated level curves together with the corresponding

computer-generated graphs. Notice that the level curves in part (c) crowd together near the

origin. That corresponds to the fact that the graph in part (d) is very steep near the origin.

z

y

z

x

x

y

(a) Level curves of f(x, y)=_xye_≈_¥

(b) Two views of f(x, y)=_xye_≈_¥

z

y

x

y

x

FIGURE 19

(c) Level curves of f(x, y)=

_3y

≈+¥+1

(d) f(x, y)=

_3y

≈+¥+1

Functions of Three or More Variables

A function of three variables, f , is a rule that assigns to each ordered triple ͑x, y, z͒ in a

domain D ʚ ‫ ޒ‬3 a unique real number denoted by f ͑x, y, z͒. For instance, the temperature

T at a point on the surface of the earth depends on the longitude x and latitude y of the point

and on the time t, so we could write T ෇ f ͑x, y, t͒.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

97817_14_ch14_p910-919.qk_97817_14_ch14_p910-919 11/8/10 1:27 PM Page 911

SECTION 14.1

FUNCTIONS OF SEVERAL VARIABLES

911

EXAMPLE 14 Find the domain of f if

f ͑x, y, z͒ ෇ ln͑z Ϫ y͒ ϩ xy sin z

SOLUTION The expression for f ͑x, y, z͒ is defined as long as z Ϫ y Ͼ 0, so the domain of

f is

D ෇ ͕͑x, y, z͒ ʦ ‫ ޒ‬3

Խ

z Ͼ y͖

This is a half-space consisting of all points that lie above the plane z ෇ y.

It’s very difficult to visualize a function f of three variables by its graph, since that would

lie in a four-dimensional space. However, we do gain some insight into f by examining its

level surfaces, which are the surfaces with equations f ͑x, y, z͒ ෇ k, where k is a constant.

If the point ͑x, y, z͒ moves along a level surface, the value of f ͑x, y, z͒ remains fixed.

z

≈+¥+z@=9

EXAMPLE 15 Find the level surfaces of the function

≈+¥+z@=4

f ͑x, y, z͒ ෇ x 2 ϩ y 2 ϩ z 2

SOLUTION The level surfaces are x 2 ϩ y 2 ϩ z 2 ෇ k, where k ജ 0. These form a family

of concentric spheres with radius sk . (See Figure 20.) Thus, as ͑x, y, z͒ varies over any

sphere with center O, the value of f ͑x, y, z͒ remains fixed.

y

x

≈+¥+z@=1

FIGURE 20

Functions of any number of variables can be considered. A function of n variables is a

rule that assigns a number z ෇ f ͑x 1, x 2 , . . . , x n ͒ to an n-tuple ͑x 1, x 2 , . . . , x n ͒ of real numbers. We denote by ‫ ޒ‬n the set of all such n-tuples. For example, if a company uses n different

ingredients in making a food product, ci is the cost per unit of the ith ingredient, and x i units

of the ith ingredient are used, then the total cost C of the ingredients is a function of the n

variables x 1, x 2 , . . . , x n :

3

C ෇ f ͑x 1, x 2 , . . . , x n ͒ ෇ c1 x 1 ϩ c2 x 2 ϩ и и и ϩ cn x n

The function f is a real-valued function whose domain is a subset of ‫ ޒ‬n. Sometimes we

will use vector notation to write such functions more compactly: If x ෇ ͗x 1, x 2 , . . . , x n ͘ , we

often write f ͑x͒ in place of f ͑x 1, x 2 , . . . , x n ͒. With this notation we can rewrite the function

defined in Equation 3 as

f ͑x͒ ෇ c ؒ x

where c ෇ ͗c1, c2 , . . . , cn ͘ and c ؒ x denotes the dot product of the vectors c and x in Vn .

In view of the one-to-one correspondence between points ͑x 1, x 2 , . . . , x n͒ in ‫ ޒ‬n and their

position vectors x ෇ ͗ x 1, x 2 , . . . , x n ͘ in Vn , we have three ways of looking at a function f

defined on a subset of ‫ ޒ‬n :

1. As a function of n real variables x 1, x 2 , . . . , x n

2. As a function of a single point variable ͑x 1, x 2 , . . . , x n ͒

3. As a function of a single vector variable x ෇ ͗x 1, x 2 , . . . , x n ͘

We will see that all three points of view are useful.

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97817_14_ch14_p910-919.qk_97817_14_ch14_p910-919 11/8/10 1:27 PM Page 912

912

PARTIAL DERIVATIVES

CHAPTER 14

14.1

Exercises

1. In Example 2 we considered the function W ෇ f ͑T, v͒, where

W is the wind-chill index, T is the actual temperature, and v is

the wind speed. A numerical representation is given in Table 1.

(a) What is the value of f ͑Ϫ15, 40͒? What is its meaning?

(b) Describe in words the meaning of the question “For what

value of v is f ͑Ϫ20, v͒ ෇ Ϫ30 ?” Then answer the question.

(c) Describe in words the meaning of the question “For what

value of T is f ͑T, 20͒ ෇ Ϫ49 ?” Then answer the question.

(d) What is the meaning of the function W ෇ f ͑Ϫ5, v͒?

Describe the behavior of this function.

(e) What is the meaning of the function W ෇ f ͑T, 50͒?

Describe the behavior of this function.

discussed in Example 3 that the production will be doubled

if both the amount of labor and the amount of capital are

doubled. Determine whether this is also true for the general

production function

P͑L, K ͒ ෇ bL␣K 1Ϫ␣

5. A model for the surface area of a human body is given by the

function

S ෇ f ͑w, h͒ ෇ 0.1091w 0.425h 0.725

where w is the weight (in pounds), h is the height (in inches),

and S is measured in square feet.

(a) Find f ͑160, 70͒ and interpret it.

(b) What is your own surface area?

2. The temperature-humidity index I (or humidex, for short) is the

perceived air temperature when the actual temperature is T and

the relative humidity is h, so we can write I ෇ f ͑T, h͒. The following table of values of I is an excerpt from a table compiled

by the National Oceanic & Atmospheric Administration.

TABLE 3

6. The wind-chill index W discussed in Example 2 has been

modeled by the following function:

W͑T, v͒ ෇ 13.12 ϩ 0.6215T Ϫ 11.37v 0.16 ϩ 0.3965Tv 0.16

Apparent temperature as a function

of temperature and humidity

Check to see how closely this model agrees with the values in

Table 1 for a few values of T and v.

Actual temperature (°F)

Relative humidity (%)

h

20

30

40

50

60

70

80

77

78

79

81

82

83

85

82

84

86

88

90

93

90

87

90

93

96

100

106

95

93

96

101

107

114

124

100

99

104

110

120

132

144

T

7. The wave heights h in the open sea depend on the speed v

of the wind and the length of time t that the wind has been

blowing at that speed. Values of the function h ෇ f ͑v, t͒ are

recorded in feet in Table 4.

(a) What is the value of f ͑40, 15͒? What is its meaning?

(b) What is the meaning of the function h ෇ f ͑30, t͒? Describe

the behavior of this function.

(c) What is the meaning of the function h ෇ f ͑v, 30͒? Describe

the behavior of this function.

TABLE 4

What is the value of f ͑95, 70͒? What is its meaning?

For what value of h is f ͑90, h͒ ෇ 100?

For what value of T is f ͑T, 50͒ ෇ 88?

What are the meanings of the functions I ෇ f ͑80, h͒

and I ෇ f ͑100, h͒? Compare the behavior of these two

functions of h.

3. A manufacturer has modeled its yearly production function P

(the monetary value of its entire production in millions of

dollars) as a Cobb-Douglas function

P͑L, K͒ ෇ 1.47L

0.65

K

0.35

where L is the number of labor hours (in thousands) and K is

the invested capital (in millions of dollars). Find P͑120, 20͒

and interpret it.

Duration (hours)

t

5

10

15

20

30

40

50

10

2

2

2

2

2

2

2

15

4

4

5

5

5

5

5

20

5

7

8

8

9

9

9

30

9

13

16

17

18

19

19

40

14

21

25

28

31

33

33

50

19

29

36

40

45

48

50

60

24

37

47

54

62

67

69

Wi nd speed (knots)

(a)

(b)

(c)

(d)

4. Verify for the Cobb-Douglas production function

8. A company makes three sizes of cardboard boxes: small,

P͑L, K ͒ ෇ 1.01L 0.75K 0.25

;

Graphing calculator or computer required

medium, and large. It costs \$2.50 to make a small box, \$4.00

1. Homework Hints available at stewartcalculus.com

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

97817_14_ch14_p910-919.qk_97817_14_ch14_p910-919 11/8/10 1:27 PM Page 913

SECTION 14.1

for a medium box, and \$4.50 for a large box. Fixed costs

are \$8000.

(a) Express the cost of making x small boxes, y medium

boxes, and z large boxes as a function of three variables:

C ෇ f ͑x, y, z͒.

(b) Find f ͑3000, 5000, 4000͒ and interpret it.

(c) What is the domain of f ?

9. Let t͑x, y͒ ෇ cos͑x ϩ 2y͒.

z

I

913

z

II

y

x

y

x

z

III

(a) Evaluate t͑2, Ϫ1͒.

(b) Find the domain of t.

(c) Find the range of t.

FUNCTIONS OF SEVERAL VARIABLES

z

IV

10. Let F ͑x, y͒ ෇ 1 ϩ s4 Ϫ y 2 .

y

(a) Evaluate F ͑3, 1͒.

(b) Find and sketch the domain of F.

(c) Find the range of F.

x

x

z

V

y

z

VI

11. Let f ͑x, y, z͒ ෇ sx ϩ sy ϩ sz ϩ ln͑4 Ϫ x 2 Ϫ y 2 Ϫ z 2 ͒.

(a) Evaluate f ͑1, 1, 1͒.

(b) Find and describe the domain of f.

12. Let t͑ x, y, z͒ ෇ x 3 y 2 zs10 Ϫ x Ϫ y Ϫ z .

(a) Evaluate t͑1, 2, 3͒.

(b) Find and describe the domain of t.

y

x

x

y

33. A contour map for a function f is shown. Use it to estimate the

values of f ͑Ϫ3, 3͒ and f ͑3, Ϫ2͒. What can you say about the

shape of the graph?

13–22 Find and sketch the domain of the function.

14. f ͑x, y͒ ෇ sxy

13. f ͑x, y͒ ෇ s2x Ϫ y

15. f ͑x, y͒ ෇ ln͑9 Ϫ x Ϫ 9y ͒

2

y

16. f ͑x, y͒ ෇ sx 2 Ϫ y 2

2

17. f ͑x, y͒ ෇ s1 Ϫ x 2 Ϫ s1 Ϫ y 2

18. f ͑x, y͒ ෇ sy ϩ s25 Ϫ x 2 Ϫ y 2

19. f ͑x, y͒ ෇

sy Ϫ x 2

1 Ϫ x2

1

0

20. f ͑x, y͒ ෇ arcsin͑x 2 ϩ y 2 Ϫ 2͒

70 60 50 40

1

30

x

20

10

21. f ͑x, y, z͒ ෇ s1 Ϫ x 2 Ϫ y 2 Ϫ z 2

22. f ͑x, y, z͒ ෇ ln͑16 Ϫ 4x 2 Ϫ 4y 2 Ϫ z 2 ͒

34. Shown is a contour map of atmospheric pressure in North

America on August 12, 2008. On the level curves (called

isobars) the pressure is indicated in millibars (mb).

(a) Estimate the pressure at C (Chicago), N (Nashville),

S (San Francisco), and V (Vancouver).

(b) At which of these locations were the winds strongest?

23–31 Sketch the graph of the function.

23. f ͑x, y͒ ෇ 1 ϩ y

24. f ͑x, y͒ ෇ 2 Ϫ x

25. f ͑x, y͒ ෇ 10 Ϫ 4x Ϫ 5y

26. f ͑x, y͒ ෇ e Ϫy

27. f ͑x, y͒ ෇ y 2 ϩ 1

28. f ͑x, y͒ ෇ 1 ϩ 2x 2 ϩ 2y 2

29. f ͑x, y͒ ෇ 9 Ϫ x 2 Ϫ 9y 2

30. f ͑x, y͒ ෇ s4x 2 ϩ y 2

1016

31. f ͑x, y͒ ෇ s4 Ϫ 4x 2 Ϫ y 2

V

1016

32. Match the function with its graph (labeled I–VI). Give reasons

1012

Խ Խ Խ Խ

(a) f ͑x, y͒ ෇ x ϩ y

1

(c) f ͑x, y͒ ෇

1 ϩ x2 ϩ y2

(e) f ͑x, y͒ ෇ ͑x Ϫ y͒2

1008

Խ Խ

(b) f ͑x, y͒ ෇ xy

S

(d) f ͑x, y͒ ෇ ͑x Ϫ y ͒

2

2 2

Խ Խ Խ Խ)

C

1004

1008

1012

N

(f ) f ͑x, y͒ ෇ sin( x ϩ y

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Tài liệu bạn tìm kiếm đã sẵn sàng tải về

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